Page 195 - Analysis and Design of Energy Geostructures

168 Analysis and Design of Energy Geostructures

The mathematical formulation of the considered problem is particularly simple
because one normal stress, one normal strain and one relevant displacement, such as
σ zz ; ε zz ; w, need to satisfy in either case the following three equations:

1. One equilibrium equation (for the case of no body forces)
5 0 ð4:69Þ
@σ zz
@z

2. One stress strain relation
1
ε zz 5 σ zz 2 α T 2 T 0 Þ ð4:70Þ
ð
E

3. One strain displacement relation
ε zz 52 @w ð4:71Þ
@z
4. Boundary conditions
In the one-dimensional case, traction boundary conditions, considering the surface
of the body as traction-free, take the following form

σ zz n z 5 0 ð4:72Þ
For displacement boundary conditions

w 5 F 3 ðHÞ ð4:73Þ

The first one-dimensional modelling of problems involving energy geostructures
has been proposed by Laloui et al. (2003), with reference to single energy piles of
length L and linear thermal expansion coefficient α subjected to a temperature varia-
tion ΔT. The same problem is considered in the following for its relevance in the
analysis and design of energy piles.
If an energy pile can deform freely, it is characterised by a thermally induced strain

ð4:74Þ
th
ε 52 αΔT
f
This thermally induced strain leads to a variation in length of the energy pile of
0 th
ð4:75Þ
ΔL 5 L 2 L 52 Lε 5 LαΔT
f
where L is the energy pile length after the application of the temperature variation
0
(cf. Fig. 4.15).
When the thermally induced deformation is completely blocked
ð4:76Þ
th th
ε 52 ε 5 αΔT
b f